FIR filter design - Continuous and Discrete Time Signals and Systems - page 665

CHAPTER

15

FIR filter design

In Chapter 14, we defined frequency-selective filters as systems that modify
the frequency components of the input signals in a predefined manner. Further
classification of frequency-selective filters is based on the length N of their
impulse responses h [ k ]. If the length N of the impulse response of a frequency-
selective filter is finite, the filter is referred to as a finite impulse response (FIR)
filter. If the length N is infinite, the frequency-selective filter is referred to as
an infinite impulse response (IIR) filter. In this chapter, we consider the design
of frequency-selective FIR filters.
The design of digital filters involves three distinct stages. Stage 1 describes
the desired specifications of the frequency characteristics of the filter. Based
on the specified frequency characteristics, stage 2 derives the transfer function
H ( z ), or the impulse response h [ k ], of the filter. Finally, stage 3 develops the
canonical realization of the filter using one of the several forms presented in
Chapter 14. While deriving the impulse response h [ k ] of an FIR filter in stage 2,
the following two conditions must also be satisfied.

(1) Causality condition. This implies that the impulse response h [ k ]ofanFIR
filter is zero for k < 0. This will ensure a causal, and hence a physically
realizable, filter.
(2) Linear-phase condition. This implies that the impulse response h [ k ]of
an FIR filter of length N is symmetrical or anti-symmetrical, i.e. h [ k ] =
h [ N − 1 − k ]. The linear-phase condition ensures that no distortion is
introduced in the input frequency components lying within the pass band
of the FIR filter.

Generally, FIR filters are designed directly from the impulse response of an
ideal lowpass filter. Section 15.1 describes the windowing approach, where an
appropriate window function w [ k ] is used to truncate the impulse response of
an ideal lowpass filter to a finite length N . The specifications of the FIR filter,
along with the characteristics of the window function, are used to calculate
the length N of the FIR filter. Sections 15.2 and 15.4 extend the windowing

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